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In this study, we focus on the stability analysis of the RLC model by employing differential equations with Hadamard fractional derivatives. We prove the existence and uniqueness of solutions using Banach's contraction principle and Schaefer's fixed point theorem. To facilitate our key conclusions, we convert the problem into an equivalent integro-differential equation. Additionally, we explore several versions of Ulam's stability findings. Two numerical examples are provided to illustrate the applications of our main results. We also observe that modifications to the Hadamard fractional derivative lead to asymmetric outcomes. The study concludes with an applied example demonstrating the existence results derived from Schaefer's fixed point theorem. These findings represent novel contributions to the literature on this topic, significantly advancing our understanding. [ABSTRACT FROM AUTHOR]

Using the coincidence degree theory of Mawhin and constructing appropriate operators, we investigate the existence of solutions to Hadamard fractional differential equations (FRDEs) at resonance ... where 1 < γ < 2, f: [1, e]×R² → R satisfies Carathéodory conditions, ∫e1 u(t)dA(t) is the Riemann-Stieltjes integration, and (HDγu) is the Hadamard fractional derivation of u of order γ. An example is included to illustrate our result. [ABSTRACT FROM AUTHOR]

Hadamard type fractional calculus involves logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenge in numerical treatment. In this paper we present a spectral collocation method with mapped Jacobi log orthogonal functions (MJLOFs) as basis functions and obtain an efficient algorithm to solve Hadamard type fractional differential equations. We develop basic approximation theory for the MJLOFs and derive a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. We demonstrate the effectiveness of the new method for the nonlinear initial and boundary problems, i.e, the fractional Helmholtz equation, and the fractional Burgers equation. [ABSTRACT FROM AUTHOR]

General variable-order fractional scale derivatives are introduced and studied. Both the stretching and the shrinking cases are considered for definitions of the derivatives of the GL type and of the Hadamard type. Their properties are deduced and discussed. Fractional variable-order systems of autoregressive–moving-average type are introduced and exemplified. The corresponding transfer functions are obtained and used to find the corresponding impulse responses. [ABSTRACT FROM AUTHOR]

Duality is one of the most interesting properties of the Laplace and Fourier transforms associated with the integer-order derivative. Here, we will generalize it for fractional derivatives and extend the results to the Mellin, Z and discrete-time Fourier transforms. The scale and nabla derivatives are used. Some consequences are described. [ABSTRACT FROM AUTHOR]

A Halanay inequality with distributed delay of non-convolution type is considered. We establish a decay of solutions as a Mittag-Leffler function composed with a logarithmic function. A general sufficient condition is found and a large class of admissible retardation kernels is provided. This needs the preparation of several lemmas on properties of the Hadamard derivative and some basic fractional differential problems with this kind of derivative. The obtained result is then applied to a Hopfield neural network system to discuss its stability. [ABSTRACT FROM AUTHOR]

This paper deals with a time-fractional Fokker–Planck equation, where the time-fractional derivative (denoted by H D a , t 1 - α u ) is in the Hadamard sense with order α ∈ (0 , 1) . With the help of the modified Laplace transform and its inverse transform, the mild solutions of the considered equation are constructed. The existence and uniqueness of the mild solutions are proved by the contraction mapping principle, and some regularity estimates are satisfied. For α ∈ (1 / 2 , 1) , the mild solution is shown to be the classical solution. The decay estimates of the solution u and H D a , t 1 - α u are also investigated. [ABSTRACT FROM AUTHOR]

The fractional scale-invariant systems are introduced and studied, using an operational formalism. It is shown that the impulse and step responses of such systems belong to the vector space generated by some special functions here introduced. For these functions, the fractional scale derivative is a decremental index operator, allowing the construction of an algebraic framework that enables to compute the impulse and step responses of such systems. The effectiveness and accuracy of the method are demonstrated through various numerical simulations. [ABSTRACT FROM AUTHOR]

A general fractional scale derivative is introduced and studied. Its relation with the Hadamard derivatives is established and reformulated. A new derivative similar to the Grünwald–Letnikov's is deduced. Tempered versions are also introduced. Scale-invariant systems are described and exemplified. For solving the corresponding differential equations, a new logarithmic Mittag-Leffler series is proposed. [ABSTRACT FROM AUTHOR]

In this manuscript, we examine both the existence, uniqueness, and the stability of solutions to the boundary value problem (BVP) of Hadamard fractional differential equations of variable order by converting it into an equivalent standard Hadamard (BVP) of the fractional constant order with the help of the generalized intervals and the piecewise constant functions. All results in this study are established using Krasnoselskii fixed‐point theorem and the Banach contraction principle. Further, the Ulam–Hyers stability of the given problem is examined, and finally, we construct an example to illustrate the validity of the observed results. [ABSTRACT FROM AUTHOR]

In this paper, a rigorous Lyapunov direct method (LDM) is proposed to analyze the stability of fractional non-linear systems involving Hadamard or Caputo–Hadamard derivatives. Based on the characteristics of Hadamard-type calculus, several new inequalities are derived for different definitions. By means of the developed inequalities and modified Laplace transform, the sufficient conditions can be derived to guarantee the Hadamard–Mittag–Leffler (HML) stability of the systems. Lastly, two illustrative examples are given to show the effectiveness of our proposed results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a notion of higher-order directional derivatives in the sense of Hadamard for set-valued maps, which is a natural extension of the classical directional derivatives. Some of the usual calculus rules, for unions, intersections, products, sums, and compositions are given under directional metric subregularity conditions. The Hadamard differentiability of the efficient value mapping and a formula to compute its derivative are also obtained. Then, we apply these derivatives to establish an implicit set-valued map theorem and employ it to higher-order sensitivity analysis of the solution mapping for a parametric vector equilibrium problem. Sensitivity for solutions to a parametric generalized equation is also investigated. Many examples are provided for analyzing and illustrating the obtained results. [ABSTRACT FROM AUTHOR]

We present new ideas to establish new results for the existence and uniqueness of solutions for a class of nonlinear sequential fractional differential systems that involve Hadamard, Riemann-Liouville and Caputo operators with initial conditions. [ABSTRACT FROM AUTHOR]

In this paper we study the existence, uniqueness and stability of solutions to the boundary value problem of Hadamard fractional differential equations of variable order. Our results are based on two fixed point theorems and Ulam–Hyers–Rassias stability definition, and can be considered as a contribution to this emerging field. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability and logarithmic decay of the solutions to fractional differential equations (FDEs). Both linear and nonlinear cases are included. And the fractional derivative is in the sense of Hadamard or Caputo–Hadamard with order α (0 < α < 1) . The solutions can be expressed by Mittag–Leffler functions through applying the modified Laplace transform. In view of the asymptotic expansions of Mittag–Leffler function, we discuss the stability and logarithmic decay of the solution to FDEs in great detail. [ABSTRACT FROM AUTHOR]

In survival or reliability studies, the mean residual life (MRL) function is an important characteristic in understanding the survival or ageing process. In this article, we consider the problem of nonparametric MRL function estimation with length-biased right-censored data. Two nonparametric estimators of the MRL are proposed and their weak convergence is presented. In order to evaluate the performance of these estimators, small Monte Carlo simulations are carried out. Results show that the proposed estimators work well especially when the sample size is small and their calculations are simple. Finally, a real data example is provided. [ABSTRACT FROM AUTHOR]

In this manuscript, we examine both the existence and the stability of solutions of the boundary value problems of Hadamard-type fractional differential equations of variable order. New outcomes are obtained in this paper based on the Darbo’s fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of the observed results.

A boundary value problem for Hadamard fractional differential equations of variable order is studied. Note the symmetry of a transformation of a system of differential equations is connected with the locally solvability which is the same as the existence of solutions. It leads to the necessity of obtaining existence criteria for a boundary value problem for Hadamard fractional differential equations of variable order. Also, the stability in the sense of Ulam–Hyers–Rassias is investigated. The results are obtained based on the Kuratowski measure of noncompactness. An example illustrates the validity of the observed results.

In this paper we obtain new estimates of the Hadamard fractional derivatives of a function at its extreme points. The extremum principle is then applied to show that the initial-boundary-value problem for linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution depends continuously on the initial and boundary conditions. The extremum principle for an elliptic equation with a fractional Hadamard derivative is also proved. [ABSTRACT FROM AUTHOR]

This article deals with some existence and uniqueness result of random solutions for some coupled systems of Hilfer and Hilfer–Hadamard fractional differential equations with random effects. Some applications are made of generalizations of classical random fixed point theorems on generalized Banach spaces.

We are concerned to establish sufficient conditions for the existence of solutions for Boundary value problem of fractional differential inclusions involving the Hadamard type derivative with order r ∈ (1, 2] in Banach spaces. [ABSTRACT FROM AUTHOR]

A Hadamard type fractional integro-differential equation on infinite intervals is considered. By using monotone iterative technique, we not only get the existence of positive solutions, but also seek the positive minimal and maximal solutions and get two explicit monotone iterative sequences which converge to the extremal solutions. At last, to illustrative the main result, an example is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the asymptotic behavior of solutions of higher order fractional differential equations with Caputo-type Hadamard derivatives of the form where r = n+α-1, α ∊ (0,1), and n ∊ℤ+. They also apply their technique to investigate the oscillatory and asymptotic behavior of solutions of the related integral equation [ABSTRACT FROM AUTHOR]

In 1923, Hadamard encountered a class of integrals with strong singularities when using a particular Green’s function to solve the cylindrical wave equation. He ignored the infinite parts of such integrals after integrating by parts. Such an idea is very practical and useful in many physical models, e.g., the crack problems of both planar and three-dimensional elasticities. In this paper, we present the rectangular and trapezoidal formulas to approximate the Hadamard derivative by the idea of the finite part integral. Then, we apply the proposed numerical methods to the differential equation with the Hadamard derivative. Finally, several numerical examples are displayed to show the effectiveness of the basic idea and technique.

In this paper, we investigate a new class of mixed initial value problems of Hadamard and Riemann-Liouville fractional integro-differential inclusions. The existence of solutions for convex valued (the upper semicontinuous) case is established by means of Krasnoselskii's fixed point theorem for multivalued maps and nonlinear alternative criterion, while the existence result for non-convex valued maps (the Lipschitz case) relies on a fixed point theorem due to Covitz and Nadler. Illustrative examples are also included. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of the unique solution for a Hadamard fractional integral boundary value problem of a Hadamard fractional integro-differential equation with the monotone iterative technique. Two examples to illustrate our result are given. [ABSTRACT FROM AUTHOR]

This paper studies the existence of solutions for a mixed initial value problem of Hadamard and Riemann–Liouville fractional integro-differential inclusions by means of endpoint theory. The main result is well illustrated with the aid of example. [ABSTRACT FROM AUTHOR]

We are concerned with an extension of a coupled sequential differential system of fractional type. Using the Banach contraction principle, we establish new results for the existence and uniqueness of solutions. Then, we prove another existence result via Schaefer's fixed point theorem. At the end, we illustrate one main result by an example. [ABSTRACT FROM AUTHOR]

In this paper, we prove existence and uniqueness results for a mixed initial value problem which contains Hadamard derivative and Riemann-Liouville fractional integrals by using standard fixed point theorems. Examples illustrating the results are also presented. [ABSTRACT FROM AUTHOR]

We obtain the Mellin transforms of the generalized fractional integrals and derivatives that generalize the Riemann–Liouville and the Hadamard fractional integrals and derivatives. We also obtain interesting results, which combine generalized δ r , m operators with generalized Stirling numbers and Lah numbers. For example, we show that δ 1 , 1 corresponds to the Stirling numbers of the 2nd kind and δ 2 , 1 corresponds to the unsigned Lah numbers. Further, we show that the two operators δ r , m and δ m , r , r , m ∈ N , generate the same sequence given by the recurrence relation. S ( n , k ) = ∑ i = 0 r m + ( m - r ) ( n - 2 ) + k - i - 1 r - i r i S ( n - 1 , k - i ) , 0 < k ⩽ n , with S ( 0 , 0 ) = 1 and S ( n , 0 ) = S ( n , k ) = 0 for n > 0 and 1 + min { r , m } ( n - 1 ) < k or k ⩽ 0 . Finally, we define a new class of sequences for r ∈ 1 3 , 1 4 , 1 5 , 1 6 , … and in turn show that δ 1 2 , 1 corresponds to the generalized Laguerre polynomials. [ABSTRACT FROM AUTHOR]

In this paper, a class of nonlinear fractional order differential impulsive systems with Hadamard derivative is discussed. First, a reasonable concept on the solutions of fractional impulsive Cauchy problems with Hadamard derivative and the corresponding fractional integral equations are established. Second, two fundamental existence results are presented by using standard fixed point methods. Finally, two examples are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

TheL-statistics form an important class of estimators in nonparametric statistics. Its members include trimmed means and sample quantiles and functions thereof. This article is devoted to theory and applications ofL-statistics for repeated measurements data, wherein the measurements on the same subject are dependent and the measurements from different subjects are independent. This article has three main goals: (a) Show that theL-statistics are asymptotically normal for repeated measurements data. (b) Present three statistical applications of this result, namely, location estimation using trimmed means, quantile estimation and construction of tolerance intervals. (c) Obtain a Bahadur representation for sample quantiles. These results are generalisations of similar results for independently and identically distributed data. The practical usefulness of these results is illustrated by analysing a real data-set involving measurement of systolic blood pressure. The properties of the proposed point and interval estimators are examined via simulation. [ABSTRACT FROM AUTHOR]

Abstract: The classical functional delta method (FDM) provides a convenient tool for deriving the asymptotic distribution of statistical functionals from the weak convergence of the respective empirical processes. However, for many interesting functionals depending on the tails of the underlying distribution this FDM cannot be applied since the method typically relies on Hadamard differentiability w.r.t. the uniform sup-norm. In this article, we present a version of the FDM which is suitable also for nonuniform sup-norms, with the outcome that the range of application of the FDM enlarges essentially. On one hand, our FDM, which we shall call the modified FDM, works for functionals that are “differentiable” in a weaker sense than Hadamard differentiability. On the other hand, it requires weak convergence of the empirical process w.r.t. a nonuniform sup-norm. The latter is not problematic since there exist strong respective results on weighted empirical processes obtained by Shorack and Wellner (1986) , Shao and Yu (1996) , Wu (2008) , and others. We illustrate the gain of the modified FDM by deriving the asymptotic distribution of plug-in estimates of popular risk measures that cannot be treated with the classical FDM. [ABSTRACT FROM AUTHOR]

Abstract: In univariate statistics, the trimmed mean has long been regarded as a robust and efficient alternative to the sample mean. A multivariate analogue calls for a notion of trimmed region around the center of the sample. Using Tukey''s depth to achieve this goal, this paper investigates two types of multivariate trimmed means obtained by averaging over the trimmed region in two different ways. For both trimmed means, conditions ensuring asymptotic normality are obtained; in this respect, one of the main features of the paper is the systematic use of Hadamard derivatives and empirical processes methods to derive the central limit theorems. Asymptotic efficiency relative to the sample mean as well as breakdown point are also studied. The results provide convincing evidence that these location estimators have nice asymptotic behavior and possess highly desirable finite-sample robustness properties; furthermore, relative to the sample mean, both of them can in some situations be highly efficient for dimensions between 2 and 10. [Copyright &y& Elsevier]

Abstract: In this paper we suggest a completely nonparametric test for the assessment of similar marginals of a multivariate distribution function. This test is based on the asymptotic normality of Mallows distance between marginals. It is also shown that the n out of n bootstrap is weakly consistent, thus providing a theoretical justification to the work in Czado, C. and Munk, A. [2001. Bootstrap methods for the nonparametric assessment of population bioequivalence and similarity of distributions. J. Statist. Comput. Simulation 68, 243–280]. The test is extended to cross-over trials and is applied to the problem of population bioequivalence, where two formulations of a drug are shown to be similar up to a tolerable limit. This approach was investigated in small samples using bootstrap techniques in Czado, C., Munk, A. [2001. Bootstrap methods for the nonparametric assessment of population bioequivalence and similarity of distributions. J. Statist. Comput. Simulation 68, 243–280], showing that the bias corrected and accelerated bootstrap yields a very accurate and powerful finite sample correction. A data example is discussed. [Copyright &y& Elsevier]

In this manuscript, we examine both the existence and the stability of solutions of the boundary value problems of Hadamard-type fractional differential equations of variable order. New outcomes are obtained in this paper based on the Darbo's fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of the observed results. [ABSTRACT FROM AUTHOR]

A boundary value problem for Hadamard fractional differential equations of variable order is studied. Note the symmetry of a transformation of a system of differential equations is connected with the locally solvability which is the same as the existence of solutions. It leads to the necessity of obtaining existence criteria for a boundary value problem for Hadamard fractional differential equations of variable order. Also, the stability in the sense of Ulam–Hyers–Rassias is investigated. The results are obtained based on the Kuratowski measure of noncompactness. An example illustrates the validity of the observed results. [ABSTRACT FROM AUTHOR]

Consider [Formula: see text] such that F ( λ , 0) = 0 for all [Formula: see text], where X and Y are Banach spaces. Bifurcation from the line [Formula: see text] of trivial solutions is investigated in cases where F ( λ , · ) need not be Fréchet differentiable at 0. The main results provide sufficient conditions for μ to be a bifurcation point and yield global information about the connected component of [Formula: see text] containing ( μ , 0). Some necessary conditions for bifurcation are also formulated. The abstract results are used to treat several singular boundary value problems for which Fréchet differentiability is not available. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

In this paper, we generalize the well-known Halanay inequality from the integer-order case to the fractional case. Namely, we consider Halanay inequalities involving Hadamard fractional derivative, and discrete and distributed delays. This inequality is then applied to study the stability of neural network systems of Hopfield type. To this end, we establish some properties like the Hadamard derivative of the product of two functions and the permutation of the Hadamard derivative and the integral or the Hadamard derivative of the convolution of two functions. It is proved that the stability is of Mittag–Leffler type, in case the distributed delay kernels themselves are Mittag–Leffler decaying to zero. [ABSTRACT FROM AUTHOR]